Events for 01/17/2019 from all calendars

Geometry Seminar

Time: 1:00PM - 2:00PM

Location: BLOC 628 **note

Speaker: Amy Hang Huang, University of Wisconsin Madison

Title: Equations of Kalman Varieties

Abstract: Given a subspace L of a vector space V, the Kalman variety consists of all matrices of V that have a nonzero eigenvector in L. I will discuss how to apply Kempf Vanishing technique with some more explicit constructions to get a long exact sequence involving coordinate ring of Kalman variety, its normalization and some other related varieties in characteristic zero. Time permitting I will also discuss how to extract more information from the long exact sequence including the minimal defining equations for Kalman varieties.

Colloquium - Wencai Liu

Time: 4:00PM - 5:00PM

Location: BLOC 220

Speaker: Wencai Liu, University of California - Irvine

Description:
Title: Universal arithmetical hierarchy of eigenfunctions for supercritical almost Mathieu operators
Abstract: The Harper's model is a tight-binding description of Bloch electrons on $\mathbb{Z}^2$ under a constant transverse magnetic field. In 1964, Mark Azbel predicted that both spectra and eigenfunctions of this model have self-similar hierarchical structure driven by the continued fraction expansion of the irrational magnetic flux. In 1976, the hierarchical structure of spectra was discovered numerically by Douglas Hofstadter, and was later observed in various experiments. The mathematical study of Harper's model led to the development of spectral theory of the almost Mathieu operator, with the solution of the Ten Martini Problem partially confirming the fractal structure of the spectrum. In this talk we will present necessary background and discuss the main ideas behind our confirmation (joint with S. Jitomirskaya) of Azbel's second prediction of the structure of the eigenfunctions. More precisely, we show that the eigenfunctions of the almost Mathieu operators in the localization regime, feature self-similarity governed by the continued fraction expansion of the frequency. These results also lead to the proof of sharp arithmetic transitions between pure point and singular continuous spectra, both in the frequency and the phase, as conjectured in 1994.

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